xleadd2d - Mathbox for Glauco Siliprandi

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Theorem xleadd2d 45786

Description: Addition of extended reals preserves the "less than or equal to" relation, in the right slot. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

**Hypotheses**
Ref	Expression
xleadd2d.1	⊢ (𝜑 → 𝐴 ∈ ℝ^*)
xleadd2d.2	⊢ (𝜑 → 𝐵 ∈ ℝ^*)
xleadd2d.3	⊢ (𝜑 → 𝐶 ∈ ℝ^*)
xleadd2d.4	⊢ (𝜑 → 𝐴 ≤ 𝐵)

**Assertion**
Ref	Expression
xleadd2d	⊢ (𝜑 → (𝐶 +_𝑒 𝐴) ≤ (𝐶 +_𝑒 𝐵))

**Proof of Theorem xleadd2d**
Step	Hyp	Ref	Expression
1		xleadd2d.1	. 2 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
2		xleadd2d.2	. 2 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
3		xleadd2d.3	. 2 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
4		xleadd2d.4	. 2 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
5		xleadd2a 13201	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ 𝐴 ≤ 𝐵) → (𝐶 +_𝑒 𝐴) ≤ (𝐶 +_𝑒 𝐵))
6	1, 2, 3, 4, 5	syl31anc 1382	1 ⊢ (𝜑 → (𝐶 +_𝑒 𝐴) ≤ (𝐶 +_𝑒 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2121 class class class wbr 5075 (class class class)co 7360 ℝ^*cxr 11173 ≤ cle 11175 +_𝑒 cxad 13056

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4842 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-id 5516 df-po 5529 df-so 5530 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-ov 7363 df-oprab 7364 df-mpo 7365 df-1st 7935 df-2nd 7936 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-xadd 13059

This theorem is referenced by: xle2addd 45795 meaunle 46921

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